Distribution of Geometrically Weighted Sum of Bernoulli Random Variables

Transcription

1 Appled Mathematc, 0,, 8-86 do:046/am095 Publhed Ole Novembe 0 ( Dtbuto of Geometcally Weghted Sum of Beoull Radom Vaable Abtact Deepeh Bhat, Phazamle Kgo, Ragaath Naayaachaya Ratthall Depatmet of Stattc, Cetal Uvety of Raatha, Khagah, Ida Depatmet of Stattc, Uvety of Botwaa, Gabooe, Botwaa E-mal: dpeh089@gmalcom, KGOSIPM@moppubbw, 5@edffmalcom Receved July 6, 0; eved Octobe 5, 0; accepted Octobe, 0 A ew cla of dtbuto ove (0,) obtaed by codeg geometcally weghted um of depedet detcally dtbuted (d) Beoull adom vaable A expeo fo the dtbuto fucto (df) deved ad ome popete ae etablhed Th cla of dtbuto clude U(0,) dtbuto Keywod: Bay Repeetato, Pobablty Ma Fucto, Dtbuto Fucto, Chaactetc Fucto Itoducto Ufom dtbuto play a mpotat ole Stattc The extece of ufom adom vaable (v) ove the teval (0,), ug B(,/) v dcated [] A a geealzato, th pape we code the followg geometcally weghted um of d Beoull v X Z, () whee Z ae d B(,p) v The emade of the pape ogazed a follow I Secto we obta the chaactetc fucto of X ad gve a tepetato fo the vaable X I Secto we deve the dtbuto fucto of X ad pove ome of t popete I Secto 4 we dcu the extece of the dety fucto I Secto 5 dtbuto of um of a fte umbe of vaable codeed ad the gaph of t pobablty ma fucto (pmf) ad dtbuto fucto (df) ae gve the Appedx The Chaactetc Fucto ad a Applcato of the Model The Chaactetc Fucto Let F() t PX t be the df of X The by the defto of X we have Ft () P XtZ0PZ0P XtZPZ X X P t( p) P t p () Hece the chaactetc fucto (cf) () t of X atfe the equato That, () t ( p) ( t ) pe t ( t ) () t ( t ) p pe t Repeatg th ad eplacg t by t each tme, we get, fo =,, () ( ) t e t t p p () The epoductve popety of the chaactetc fucto exhbted by () compaable to the chaactetc fucto of a ftely dvble dtbuto Fo detal oe may efe to Secto 7 of [] Sce ( t ) a we have t (4) () t p pe Note that f p = 0, th fte poduct, ad, f p =, the fte poduct e t Thu p = 0 eult X beg degeeate at 0 whle p = mple that X degeeate at If p = /, the the poduct tem (4) t t e e t e t a Thu f p = / the X ha U(0,) dtbuto A Applcato Th eultg dtbuto ca be ued a a model a Copyght 0 ScRe

2 D BHATI ET AL 8 tuato mla to the followg Suppoe a patcle ha lea movemet o the teval [0,] To captue the patcle uppoe the followg bay captug techque of dvdg the extg teval to two equal halve ued Suppoe tally thee ae two bae put at 0 ad Afte oe ut of tme a bae put at the mdpot of 0 ad Futhe the teval whch the patcle foud dvded to two equal halve by placg a bae at the mdpot ad the poce cotued The teval cotag the patcle eep o hg ad fally h to X, the pot at whch the patcle captued the log u The behavo of the patcle ow oly to the extet that at the momet of placg a bae afte exactly oe ut of tme the patcle o the ght de of the eted bae wth pobablty p Ma Reult Notato It ow that evey umbe t, 0t ha a bay epeetato though a, a 0 o, a t a If a umbe t ha the epeetato t a, a, we efe t a a fte bay tem atg umbe ad uch a umbe ca be epeeted by, fo ome,,, Howeve uch a umbe alo ca be epeeted a b, b a,,,,, b 0, b,,, It to be oted that the ght tal of the equece {a } of the fom (0,0,0 ) whle that of the equece {b } of the fom (,,, ) I the followg a a matte of coveto we do ot code epeetato wth the ght tal of the fom (,,,) Ude uch a coveto, t [0,] coepod to a uque bay equece a ad coveely If deote th elato a epeetato) Popete Theoem : Let t a ad t a, the we hall t a, ( to mea the bay a,,,, 0 The 0 the df of X defed () gve by Ft () P Xt a q p (5) Poof: Let, t a ad a be the th o-zeo elemet the equece a,,, Let t be the umbe havg the bay epeetato ( a, a, a,0,0, ) ad t0 0 It to be oted that t a fte bay tematg umbe If t ot a fte bay tematg umbe the the equece t ceae to t If t a fte bay tematg umbe the we ote that t t fo ome fte Fo example, let t have a bay epeetato ad 4, 7, 8, o that t (000000), 4 t ( ), o o We ote that a othewe Note that P X t t ( ) ad fo,,, ad 0 P Z a,,,,, Z, Z 0,,, P Z, Z 0,,, 0 Thu F doe ot have a ump at t ad F t Ft Let t be ot a fte bay epeetg umbe We ote that 0 X t PZ 0,, Z q P 0 P t X t P Z 0,, Z 0, Z, Z 0,, 0 Z q p PtXt PZ0,, Z 0, Z, Z 0,, 0,, 0,, 0 Z Z Z Z q p Thu geeal fo =,,, we have Hece P t X t q p ( ) PX ( t) Pt Xt q p ( ) If we let, the a, the we wll have I fact F doe ot have ump at at t, (0 < t < ) Hece PX ( t) a q P X t P X t a q p p Copyght 0 ScRe

3 84 D BHATI ET AL Howeve, f t a fte bay temato umbe the t t fo ome fte umbe ad ce t PX tpx t P X P X t a q p a q p a 0,,, Patcula Cae If p, the PX t a t, t (0,) Hece X ~ U (0,) Rema: I Fa ct t ca be vefed that (5) atfe () It folfact that f t ha bay epeetato a = low fom the (a, a,) the fo ) 0 < t <, t/ ha bay epeetato u( u, u, ) whee u = 0 ad u + = a, =,, ad ) / < t <, t ha bay epeetato v = v, v, whee v = a +, =,, Theoem : If u ad v have bay epeetato (a, a,, a, 0, 0,,0) ad (a, a,,a,,,,) epectvely the the codtoal dtbuto of X gve u X v that of u X Poof: Follow by the defto of X Theoem : Let t a wth,,,,0,0, The fo 0 wth a a a a P 0 X t 0 X t P0 X Poof: P 0 X t 0 X t P 0 X t P0 X t 0t t P 0 X t PZ a, Z a,, Z a, Z P Z a, Z a,, Z a PZ a, Za,, ZaP Z 0 P Z a, Z a,, Z a P X PX ad Theoem 4: Fo 0 t we have q Pt X Pt X p Poof: () Z Z, Z Z, Z, Z, Let () 0, () () PZ 0PZ A qpz A P t X P Z Z A () Pt X P Z Z A P Z () () P Z A pp Z A () X I the above fact PZ A P t Hece the eult Theoem 5: If F(, p ) ad G(, p ) ae the dtbuto fucto of X a d -X epectvely the Gxp (, ) Fx (, p) Poof: If X Z ad Z ae d B (, p) the X Z Y whee Y ae d B(, p) Hece the eult Mea ad Vaace of X: E X ( ) Hece E( X) EZ p p, EZ E( ZZ ) p p p p p p( p) p p( p) Va( X ) By ug the cf t the cu- mulat ca alo be obtaed () t p pe Copyght 0 ScRe

4 D BHATI ET AL 85 4 Noextece of Dety Fucto We have poved that the dtbuto fucto of X gve by, t a Ft () a q p Let the left devatve ad the ght devatve of F at t ext Thee be deoted by f ( t ) ad f ( t ) epectvely Code f ad f, the ght ad left devatve of F at / ad F F f lm qp lm pq lm( q)( p) F F f lm lm lm( p)( q) Note that f ad f ae equal f ad oly f p q Hece F ot dffeetable at / f p Let D :,,, 0,,,, It ca be how that F ot dffeetable at each pot of D ad the et D coutable dee ubet of [0,] Hece F owhee dffeetable [0,] 5 Dtbuto of Sum of a Fte Numbe of Beoull Radom Vaable ove, the dety fucto of X doe ot ext o the teval (0,) Hece the follow g we code the e- quece {X } of v defed by X Z (6) The equece X ceae pot we to X ad X X Smla to (5), t ca be how that fo t a, () ( ) the df of X F t a q p F, whee a Futhe, at thee value of, F () F() ad the dffeece betwee two ucceve value of, a uch the two fucto F () t ad F() t ae almot ale Sce the equece {X } ceae pot we t o X the equece {F (t)} deceae to F(t) fo t (0,) We ote that fo p = /, F () t F() The gaph of pmf ad df of X 5 ad X 0 fo dffeet value of p ae gve the Appedx 6 Acowledgemet We ae thaful to Pofeo H J Vama, Cetal Uvety of Raatha, Ida, fo the dcuo whch helped to mpove the cotet ad the peetato of the pape 7 Refeece [] S Kute ad R N Ratthall, Ufom Radom Va- Academc Pe, Cambdge, able Do They Ext Subectve See? Calcutta Stattcal Aocato Bullet, Vol 4, 99, pp 4-8 [] K L Chug, A Coue Pobablty Theoy, d Edto, 00 Sce F(t) a fte ee fo t ot D, the exact evalu ato of F(t) ot poble fo each t (0,) Moe- Copyght 0 ScRe

5 86 Appedx D BHATI ET AL The gaph of pmf ad df of X fo dffeet value of p Pobablty ma fucto Dtbuto fucto = 5 = 0 = 5 = 0 p = 0 p = 0 p = 07 p = 07 p = 09 p = 09 Copyght 0 ScRe